Lasse Rempe: Papers and Preprints

Papers and Preprints

(as of July 2010)

You can see my publications listed

on MathSciNet (you will have to be connecting from an institution which has subscribed to MathSciNet),
on Zentralblatt (you will need to connect from a subscribing institution; otherwise, you will only see the three most recent publications), or
on the Preprint ArXiV.

Theses

Dynamics of Exponential Maps , doctoral thesis, Christian-Albrechts-Universität Kiel, May 2003.
(Also available on the Dynamical Systems Thesis Server at the SUNY Stony Brook .
A Limit Problem for Online Scheduling , Rapport de Stage, DEA Algorithmique, Universite de Paris-Sud, 2002.

Books

With Rebecca Waldecker: Primzahltests für Einsteiger (German); Vieweg+Teubner, 2009. Front matter, back matter and extracts from Sections 1.1 and 2.1.

Articles Accepted or Submitted for Publication

On a Question of Herman, Baker and Rippon Concerning Siegel Disks , Bull. London Math. Soc. 36 (2004), no. 4, 516 - 518. Published version.
A Landing Theorem for Periodic Rays of Exponential Maps , Proc. Amer. Math. Soc. 134 (2006), no. 9, 2639 - 2648. Published version.
Topological Dynamics of Exponential Maps on their Escaping Sets , Ergodic Theory Dynam. Systems 26 (2006), no. 6, 1939 - 1975. arXiv:math.DS/0309107; Published version.
On nonlanding dynamic rays of exponential maps, Ann. Acad. Sci. Fenn. 32 (2007), 353 - 369. Published version.
On a question of Eremenko concerning escaping components of entire functions, Bull. London Math. Soc. 39 (2007),661 - 666. Published version.
(with Markus Förster and Dierk Schleicher) Classification of Escaping Exponential Maps, Proc. Amer. Math. Soc. 136 (2008), 651 - 663. Published version.
(with Dierk Schleicher) Combinatorics of Bifurcations in Exponential Parameter Space, in Transcendental Dynamics and Complex Analysis (P. Rippon and G. Stallard, eds), Cambridge Univ. Press, 2008, 317 - 370.
Prime Ends and Local Connectivity , , Bull. London Math. Soc. 40 (2008), no. 4, 817 - 826. Published version: Abstract, PDF.
(with Dierk Schleicher) Bifurcation loci of exponential maps and quadratic polynomials: local connectivity, triviality of fibers, and density of hyperbolicity, in: Holomorphic Dynamics and Renormalization, in Honour of John Milnor's 75th birthday (M. Lyubich and M. Yampolsky, eds), Fields Institute Communications 53 (2008).
Siegel Disks and Periodic Rays of Entire Functions, J. Reine Angew. Math., 624 (2008), 81 - 102. Published version.
(with Dierk Schleicher) Bifurcations in the Space of Exponential Maps, Invent. Math. 175 (2009), no. 1, 103 - 135. Published version.
Hyperbolic dimension and radial Julia sets of transcendental functions, Proc. Amer. Math. Soc. 137 (2009), 1411-1420. arXiv:0712.4267; published version.
Rigidity of escaping dynamics for transcendental entire functions, Acta Math. 203 (2009), no. 2, 235-267. Published version.
The escaping set of the exponential, Ergodic Theory Dynam. Sytems 30 (2010) 505-599. Published version.
(with Gwyneth Stallard) Hausdorff dimension of escaping sets of transcendental entire functions, Proc. Amer. Math. Soc 138 (2010), no. 5, 1657-1665. Published version.
(with Mikhail Lyubich and Jeremy Kahn) A note on hyperbolic leaves and wild laminations of rational functions, J. Difference Equ. Appl., 16 (2010), no. 5-6, 655-665. Published version.
(with Phil Rippon and Gwyneth Stallard) Are Devaney hairs fast escaping?, J. Difference Equ. Appl., 16 (2010), no. 5-6, 739-762. Published version.
(with Günter Rottenfußer, Johannes Rückert and Dierk Schleicher) Dynamic rays of bounded-type entire functions, to appear in Ann. Math.
(with Sebastian van Strien) Absence of line fields and Mañé's theorem for non-recurrent transcendental functions, to appear in Trans. Amer. Math. Soc.
Connected escaping sets of exponential maps, to appear in Ann. Acad. Sci. Fenn.
(with Sebastian van Strien) Density of hyperbolicity for classes of real transcendental entire functions and circle maps, Preprint (2010), submitted.

Manuscripts in preparation

(with Phil Rippon) Exotic Baker and wandering domains of Ahlfors islands functions.
(with Helena Mihaljevic-Brandt) Absence of wandering domains for entire functions with escaping singular sets.
(with Xavier Buff, Arnaud Chéritat and Hiroyuki Inou) Arithmetical hedgehogs.
(with Anna Benini) Rational fibers in exponential parameter space.
(with Krzysztof Baranski and Xavier Jarque) Brushing hairs for hyperbolic entire maps of finite order
Approximation by functions in the Eremenko-Lyubich class.

Summaries and errata

The goal of this section is to make available some summaries of what is contained in the papers, aimed at other experts in the field, as well as to collect misprints and errata. If you find any of the latter, please e-mail me and I will include them here!

This information is currently very incomplete, but I hope I will add to it as time goes by, and in particular as new papers are completed.

Topological dynamics of exponential maps on their escaping sets

Ergodic Theory Dynam. Systems 26 (2006), no. 6, 1939 - 1975. arXiv:math.DS/0309107; published version.

Errata:

In the proof of Theorem 1.1, on page 1950, the number R should be defined as R:=exp(Q(K)+1)+K, instead of R:=Q(K)+1. (Many thanks to Anup Raja Lamichhane for pointing out this error.)

Hyperbolic dimension and radial Julia sets of transcendental functions

Proc. Amer. Math. Soc. 137 (2009), 1411-1420. arXiv:0712.4267; published version.

Errata:

The reference [O] should have listed Adam Epstein and Richard Oudkerk as authors, and should have been denoted by [EO].

Dynamic rays of bounded-type entire functions

With Günter Rottenfußer, Johannes Rückert and Dierk Schleicher.

Ann. of Math. 173 (2011), no. 1, 77-125. arxiv:0704.3213; published version.

Errata:

In part (c) of the statement of Proposition 5.4, M should be M' (twice).
In the statement of Lemma 5.7 (Linear head-start is preserved by composition), "F_1 has bounded slope and all F_i satisfy uniform linear head-start conditions" should be replaced by "All F_i have bounded slope and satisfy uniform linear head-start conditions." The second paragraph of the proof should be replaced by the following:

Let $α$ and $β$ be such that $F_{i} \in ℬ_{\log}^{n} (α, β)$ for all $i$ . For $i = 1, \dots, n$ , let $K_{i}$ and $M_{i}'$ be the constants from Proposition 5.4 (c), applied to $F_{i}$ . We set $K := \max_{i} K_{i}$ and $M := \max {δ, \max_{i} M_{i}'}$ , where $δ = δ (α, β, K, 0)$ is the constant from Lemma 5.2. Now fix $i$ and let $T$ be a tract of $F_{i}$ . Let $w, z \in T$ such that $Re w > K Re z + M$ and such that $F_{i} (z)$ and $F_{i} (w)$ belong to the same tract of $F_{i + 1}$ (where we use the convention that $F_{n + 1} = F_{1}$ ). Then $| w - z | \geq Re w - Re z > M \geq δ$ , and Lemma 5.2 gives that

$Re F_{i} (w) > K Re F_{i} (z) + M or Re F_{i} (z) > K Re F_{i} (w) + M .$ ()

By Proposition 5.4 (c), the first inequality must hold. It follows that $G_{a}$ satisfies a uniform linear head-start condition with constants $K$ and $M$ .

(Many thanks to Sebastian Vogel for pointing out this error.)

Exotic Baker and wandering domains of Ahlfors islands functions

With Phil Rippon.

This paper constructs examples of functions satisfying Adam Epstein's "Ahlfors islands property" (see "Hyperbolic dimension and radial Julia sets", above) which have various interesting dynamical or function-theoretic properties, using approximation theory.

If X is the Riemann sphere or a torus, and W is a proper subdomain of X, then we show that there are Ahlfors islands maps g:W->X. Moreover, these can be chosen such that:

a) g has a Baker domain whose set of limit functions coincides with any prescribed compact connected subset of the boundary of W that can be written as the accumulation set of a curve in W; moreover the iterates in the Baker domain can be chosen to tend to infinity arbitrarily slowly;
b) g has a wandering domain whose set of limit functions coincides with any prescribed compact subset of the boundary of W;
c) given a prescribed C1 curve gamma tending to the boundary, g can be constructed with a logarithmic asymptotic value that has gamma as an asymptotic curve.

Note that c) is a nondynamical statement, and in fact W can be replaced by any proper subdomain of any compact Riemann surface Y. In particular, it follows that there are no restrictions on the possible domains of Ahlfors islands functions when the target Riemann surface has genus at most 1.

While most of the results are well-known when X is the Riemann sphere and W is the complex plane or the punctured plane, the statement about escape speeds in a) also yields an apparently new result for entire functions: The rate of escape in a Baker domain of a transcendental entire function can be arbitrarily slow.

The construction uses Arakelian's approximation theorem and its generalization to analytic functions on Riemann surfaces due to Scheinberg. In order to be able to construct Baker domains, we need to establish approximation results for functions taking values in a simply-connected domain that are a priori stronger than those provided by the Arakelian-Scheinberg theorem.

We also construct examples as in a), b) and c) above when X is a compact hyperbolic Riemann surface. However, in this case the constructed function will be an Ahlfors islands map g:W'->X with the stated properties, where W' is a proper subdomain of W.